§5 Currents after closing and opening the circuit

Consider a circuit consisting of a source current EMF ?, the resistance R, inductance L and the key K. The circuit current flows

At time t = 0 disable EMF, closing the circuit shorted (key in position 2). As soon as the current begins to decrease self-induced emf arises that prevents this decrease.
Current in the circuit satisfies Ohm's Law:

Potentiation

At t = 0

 -time constant determines the rate of decrease (increase) in the circuit

therefore, τ - the time during which the current in the circuit is reduced by a factor e. The more or less L R (ie, the greater τ), the slower the current in the circuit decreases.
2. Current at closure.

, in the chain, except EMF E will act self-induced emf, which prevents the instantaneous current rise.
Then Ohm's law can be written as:

or

inhomogeneous linear differential equation.

His solution is the total solution of the homogeneous differential equation  and any particular solution of the inhomogeneous equations.

Particular solution of the nonhomogeneous

The general solution of the nonhomogeneous

at t = 0 I = 0, hence , I₀ = const,then

§6 Mutual induction.

Transformers

Consider two fixed circuit (1 and 2), located close enough to each other. If the loop current flows one I1, then the magnetic flux generated by this current is proportional to I1. We denote the part of the Ф₂₁ thread that runs through circuit 2. Then

where L₂₁ - proportionality factor .

If the current I₁is changed, the circuit 2 is induced emf E_{i2 2}

Similarly, during the flow of current in the circuit 2 I2 magnetic flux pervades circuit - Ф₁₂

If I₂is changed, then

The phenomenon emergence the EMF in a circuit when changing current in the other is called mutual induction. Proportionality coefficients L₁₂ and L₂₁are called mutual inductance

L₁₂ = L₂₁.

L₁₂ and L₂₁are dependent on the geometric shape, size, relative position of the contours and the magnetic permeability of the medium surrounding contours

[L₁₂] =Гн.

Calculate the mutual inductance of two coils wound on a common toroid. Magnetic induction field generated by the coil 1 to the number of turns N₁,the magnetic permeability μ core

where  is the length of the core in the midline

The magnetic flux through one turn of coil 2: :

The total magnetic flux

Consequently, the

2. Transformers (converter) - a device to increase, decrease, electrical isolation, etc. voltage and current.

R₁ -the resistance of the primary winding ,

I₁R₁ small in comparison  and  in variable fields. Consequently,

EMF mutual induction in the secondary

 Consequently,

Sign “ - “means that the phases  and opposite .

 - transformation ratio.

k > 1- step-up transformer ;

k < 1- step-down transformer.

Therefore, the currents in the windings are inversely proportional to the number of turns in these coils.

.